   Digital topology [1]} has been developing and increasing its scientific importance. Many significant invariants of topology, especially homotopy, homology and cohomology, have a substantial value for digital images. The digital homotopy is, in particular, our fundamental equipment. You can easily have the comprehensive knowledge about the digital homotopy from [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. As we mentioned, we are concerned with topological interpretations of robot motions in digital images. Farber [9]} studies topological complexity of motion planning. The construction of motion planning algorithms on a lot of topological spaces has been discussed. Different types of topological structures have been expertly placed in the theory of this subject [10]}. What we add to these studies is related to the part of the notion of higher topological complexity \(TC_{n}\)  in digital images. Rudyak [11]} first define it for ordinary topological spaces. It is integrated into digital topology by İs and Karaca [12]}. The digital meaning of the Lusternik-Schnirelmann category \(cat\)  appears in [13]} and the importance of the notion is its close relationship with \(TC_{2}\) , in other saying the special case of \(TC_{n}\) , when \(n=2\) . Therefore, we frequently use \(cat\)  having precious results about not only \(TC_{2}\)  but also \(TC_{n}\) . The definition of \(cat\) , \(TC_{2}\)  and \(TC_{n}\)  is expressed by the concept of the digital Schwarz genus of some digital fibrations. In a way, we can figure out that these concepts are in a close relation with all the properties of the notion digital Schwarz genus and each other. Moreover, the structure that helps us stating one of the strongest relationship between \(cat\)  and \(TC\)  is the \(\kappa -\) topological group of digital images, where \(\kappa \)  is an adjacency relation of a digital image. We introduce this notion and outline the framework of it.
